How do you find f'(3) using the limit definition given #f(x)= x^2 -5x + 3#?
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To find ( f'(3) ) using the limit definition, where ( f(x) = x^2 - 5x + 3 ), you use the formula for the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
Substitute ( x = 3 ) into the formula:
[ f'(3) = \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h} ]
[ f'(3) = \lim_{h \to 0} \frac{(3 + h)^2 - 5(3 + h) + 3 - (3^2 - 5(3) + 3)}{h} ]
[ f'(3) = \lim_{h \to 0} \frac{(9 + 6h + h^2) - 15 - 5h + 3 - (9 - 15 + 3)}{h} ]
[ f'(3) = \lim_{h \to 0} \frac{h^2 + 6h - 12}{h} ]
[ f'(3) = \lim_{h \to 0} \frac{h(h + 6)}{h} ]
[ f'(3) = \lim_{h \to 0} (h + 6) ]
[ f'(3) = 6 ]
So, ( f'(3) = 6 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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